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c. The reflexive closure of a quasi order is a partial order.

d. Every finite poset has a minimal element and a maximal element

10. List the ordered pairs in the relation R from A = {0,1, 2, 3} to B = {0, 1, 2, 3, 4} where (a, b) R if and only if

a. a>b.

b. a + b = 3.

c. a divides b.

d. a - b = 0.

e. gcd( a , b ) = 1.

f. lcm( a , b ) = 6.

11. Recursively define the relation { ( a , b ) | a=2b }, where a and b are natural numbers.

12. List unary relation on {1, 2, 3}.

13. Prove that there are 2 n2 binary relations on a set of cardinality n .

14. For each of the following relations on the set {1, 2, 3, 4}, decide whether it is reflexive, symmetric, antisymmetric and/or transitive.

a. {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)}

b. {(1, 3), (1, 4), (2, 3), (3, 4)}

c. {(1, 1), (1, 3), (1, 4), (2, 1), (2, 3), (2, 4), (3, 1), (3, 3), (3, 4)}

15. Determine whether the relation R on the set of all integers is reflexive, symmetric, antisymmetric, and/or transitive, where ( x , y ) ∈ R if and only if

a. x is divisible by y .

b. x y.

c. y = x + 2 or y = x - 2.

d. x = y ² + 1 .

16. Let A be the set of people in your town. Let R 1 be the unary relation representing the people in your town who were registered in the last election  and R 2 be the unary relation representing the people in your town who voted in the last election.  Describe the 1-tuples in each of the following relations.

a. R 1 ∪ R 2.

b. R 1 ∩ R 2.

17. Draw the directed graph that represents the relation {( a , b ), ( a , c ), ( b , c ), ( c , b ), ( c , c ), ( c , d ), ( d , a ), ( d , b )}.

18. Let R be the parent-child relation on the set of people that is, R = { ( a, b ) | a is a parent of b }.  Let S be the sibling relation on the set of people that is, R = { ( a , b ) | a and b are siblings (brothers or sisters) }.  What are S o R and R o S ?

19. Let R be a reflexive relation on a set A .  Show that R n is reflexive for all positive integers n .

20. Let R be the relation on the set { 1, 2, 3, 4} containing the ordered pairs (1, 1), (1, 2), (2, 2), (2, 4), (3, 4), and (4, 1).  Find

a. the reflexive closure of R

b. symmetric closure of R   and

c. transitive closure of R .

21. Let R be the relation { ( a, b ) | a is a (integer) multiple of b } on the set of integers.  What is the symmetric closure of R ?

22. Suppose that a binary relation R on a set A is reflexive.  Show that   R*   is reflexive, where   R* = i = 1 n R i size 12{ union rSub { size 8{i=1} } rSup { size 8{n} } {R} rSup { size 8{i} } } {} .

23. Which of the following relations on {1, 2, 3, 4} are equivalence relations?  Determine the properties of an equivalence relation that the others lack.

a. {(1, 1), (2, 2), (3, 3), (4, 4)}

b. {(1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4, 4)}

c. {(1, 1), (1, 2), (1, 4), (2, 2), (2, 4), (3, 3), (4, 1), (4, 2), (4, 4)}

24. Suppose that A is a nonempty set, and f is a function that has A as its domain.  Let R be the relation on A consisting of all ordered pairs ( x , y ) where f(x) = f(y) .

a. Show that R is an equivalence relation on A .

b. What are the equivalence classes of R ?

25. Show that propositional equivalence is an equivalence relation on the set of all compound propositions.

26. Give a description of each of the congruence classes modulo 6.

27. Which of the following collections of subsets are partitions of {1, 2, 3, 4, 5, 6}?

a. {1, 2, 3}, {3, 4}, {4, 5, 6}

b. {1, 2, 6}, {3, 5}, {4}

c. {2, 4, 6}, {1, 5}

d. {1, 4, 5}, {2, 3, 6}

28. Consider the equivalence relation on the set of integers R = { ( x, y ) | x - y is an integer}.

a. What is the equivalence class of 1 for this equivalence relation?

b. What is the equivalence class of 0.3 for this equivalence relation?

29. Which of the following are posets?

a. ( Z,  =  )

b. ( Z, ≠)

c. ( A collection of sets, ⊆).

30. Draw the Hasse diagram for the divisibility relation on the following sets

a. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

b. {1, 2, 5, 8, 16, 32}

31. Answer the following questions concerning the poset ({{1}, {2}, {3}, {4}, {1, 3}, {1, 4}, {2, 4}, {3, 4}, {1, 2, 4}, {2, 3, 4}},⊆).

a. Find the maximal elements.

b. Find the minimal elements.

c. Is there a greatest element?

d Is there a least element?

e. Find all upper bounds of {{2}, {4}}.

f. Find the least upper bound of {{2}, {4}}, if it exists.

g. Find all lower bounds of {{1, 2, 4}, {2, 3, 4}}

h. Find the greatest lower bound of {{1, 2, 4}, {2, 3, 4}}, if it exists.

Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Source:  OpenStax, Discrete structures. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10768/1.1
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